Maximum Matching in the Online Batch-Arrival Model - PowerPoint Presentation

Maximum Matching in the Online Batch-Arrival Model
Maximum Matching in the Online Batch-Arrival Model

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Sahil Singla Carnegie Mellon University Joint work with Euiwoong Lee 26 th June 2017 TwoStage matching problem Graph Edges Appears in Two Batches Stages Appears in Stage 1 ID: 611388 Download Presentation

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Slide1

Maximum Matching in the Online Batch-Arrival Model

Sahil Singla (Carnegie Mellon University)Joint work with Euiwoong Lee

26th June, 2017Slide2

Two-Stage matching problem

Graph Edges Appears in Two Batches/ Stages

Appears in Stage 1

Pick Matching

in (Unknown )Unselected Edges Disappear Appears in Stage 2Select in s.t. is a MatchingGoalMaximize size of Competitive Ratio:Greedy is Half Competitive

 

2

 

Can we beat half?Slide3

The Z Graph

Graph Appears in Two Batches

Appears

Pick Matching

in

(Unknown )Unselected Edges Disappear AppearsSelect in s.t. is a MatchingGoalMaximize size of

 3

 

 

 

Do we Pick Edge in

?

Pick

w.p

.

Case 1

: E[

Alg

]=

& OPT=1

Case 2

:

E[

Alg

]=

&

OPT=2

Fractional Matching?

Easier than Integral

 

or

Case 1

Case 2Slide4

Our results

Theorem 1

:

For

Two-Stage Integral Bipartite

Matching, There Exists a Competitive Tight Algorithm. 4

Theorem 2: For

Two-Stage Fractional Bipartite Matching, There Exists an Instance Optimal

Competitive Algorithm.Instance Optimal: Given returns s.t. Gets for every For every Alg,

where ALG

 Slide5

Prior Work

5Online ArrivalSingle arrival in each step (linear # stages)Immediate & Irrevocable decisionsVertex Arrival or Edge ArrivalSemi-Streaming Arrival

decisions postponed

Vertex Arrival or Edge Arrival

Two-Stage Stochastic Optimization

Costs change every stageArrival from a known distribution Slide6

OUTLINE

Multi-Stage MatchingExamples & Special CasesProof Idea: Fractional Bipartite Matching

Proof Idea: Integral

Bipartite

Matching

Extensions and Open Problems6Slide7

Randomly Pick Max Matching?

7Find a Max Matching in

Pick it Randomly, and

Nothing

Otherwise

What if Multiple Max Matchings? Which one to pick?With how much probability? Graphs Known Where For Every Max Matching , Randomly Picking gives  

 

 Slide8

has A Perfect Matching

 

Suppose

has a Perfect Matching M

Every vertex with an incident

edge in is matched in MPick M w.p. , and Nothing OtherwiseOptimally Augment in Stage 2How to Prove ? 8

Lemma

:

Above algorithm is Competitive for Two-Stage Integral Bipartite Matching. Slide9

Primal-Dual Framework

9Offline Bipartite Matching LP

 

 

max

 

min

 

s.t.

s.t.

 

 

Opt Solution Certificate For

Show

feasible

s.t.

-Approx Solution Certificate For

Show

-feasible

s.t

.

 

i.e.,

 Slide10

has A Perfect Matching

 

ALGORITHMPick M w.p

.

, & Optimally Augment in Stage 2Set = 1 when is picked in Stage 1Set = 1 when is picked in Stage 2 Set  

10

Lemma

: Above algorithm is

Competitive.

 

Certificate:

s.t.

&

Set

when

matched

in Stage 1

Set

to be

optimal vertex

cover

for Stage 2, where

Set

 Slide11

has A Perfect Matching

 Analysis

: Since

-Feasibility

: Case analysis ,

Both

in

:

Both not in

:

Only

in

:

 

11

Q.E.D.Slide12

OUTLINE

Multi-Stage MatchingExamples & Special CasesProof Idea: Fractional Bipartite Matching

Proof Idea: Integral

Bipartite

Matching

Extensions and Open Problems12Slide13

Two-Stage Fractional Matching

13

Theorem 2

:

For

Two-Stage Fractional Bipartite Matching, There Exists an Instance Optimal Competitive Algorithm.Proof Idea:

Construct an LP on

that maximizes

Gets for every For every ALG, where ALG

 

Here

OPT(

)

 Slide14

A New LP

14

 

max

s.t

.

 

 

 

 

Instance Optimality:

Gets

for every

For

every

ALG

,

where

ALG

 

Ques:

Is

?

 

 

Let

 Slide15

OUTLINE

Multi-Stage MatchingExamples & Special CasesProof Idea: Fractional Bipartite Matching

Proof Idea: Integral

Bipartite

Matching

Extensions and Open Problems15Slide16

is Expanding 

Suppose

is

ExpandingEvery has neighborsCan pick a random matching s.t. & we have &

 

16

  

Here

 

Algorithm

Pick

M

w.p

.

, & Optimally Augment in

Stage 2

Analysis

Set

&

for

when

picked

For any

case-by-case

show for every edge

 Slide17

Two-Stage Integral Matching

17

Theorem 1

:

For

Two-Stage Integral Bipartite Matching, There Exists a Competitive Tight Algorithm. 

Algorithm

:

Construct a Matching Skeleton of Partition into several Expanding Bipartite SubgraphsRandomly Pick a Max Matching in each Bipartite SubgraphOptimally Augment in Stage 2Proof: Show s.t.

where

for every edge

 Slide18

Bipartite Matching Skeleton

18Algorithm :Construct a Matching Skeleton of

Randomly pick a Max Matching in each bipartite subgraph

Optimally

augment

in Stage 2 Goel-Kapralov-KhannaDecompose into is expanding, where

No edge

to

No edge to for Algorithm

Select

uniformly

pick

if

Analysis

Set

&

for

For

any

show for every edge

 

 

 

 

 

 

 

 

 

 

 

 Slide19

OUTLINE

Multi-Stage MatchingExamples & Special CasesProof Idea: Fractional Bipartite Matching

Proof Idea: Integral Bipartite

Matching

Extensions and Open Problems

19Slide20

Extensions

Theorem 3

:

For

Two-Stage Fractional General

Matching, There Exists a Competitive Algorithm. 20

Theorem 4

: For s-Stage Integral General Matching

, There Exists a Competitive Algorithm. Slide21

General Matching Skeleton

21Edmonds-Gallai Decomposition

Proof Idea:

Run

Bipartite

Algo for Pick Matching in synchronously with Distribute

duals to

vertices & odd-components

Show for any : for every

 

has

vertex from each odd component

 Slide22

Open Problems

22

Problem 1:

For

s-Stage Integral Bipartite Matching, Does There Exist an Algorithm That Beats Half by a Constant?Problem 2: For Two-Stage Integral General Matching, What is the Tight Competitive Ratio?

We showed it’s

and

 Slide23

Open Problems

23

Problem 3:

Any Natural Online Problem With

Competitive Algorithm in s-Stage Online-Batch Arrival Model? 

Not True For

Online Set Cover

Online Facility LocationOnline Steiner TreeUnrelated Load Balancing (makespan minimization)Slide24

summary

Fractional Bipartite MatchingInstance optimal for two stagesIntegral Bipartite Matching competitive for two-stages

Integral General Matching

competitive

for

s-stage sOpen ProblemsBeat half for linear # stages?Other interesting multistage problems? 24Questions?Slide25

references

L. Epstein, A. Levin, D. Segev, and O. Weimann. Improved bounds for online preemptive matching. STACS’13A. Goel, M. Kapralov, and S. Khanna. `On the communication and streaming complexity of maximum bipartite

matching’. SODA’12D. Golovin, V. Goyal, V.

Polishchuk

, R. Ravi, and M.

Sysikaski. `Improved approximations for two-stage min-cut and shortest path problems under uncertainty’. Math Prog’15R. M. Karp, U. V. Vazirani, and V. V. Vazirani. `An optimal algorithm for on-line bipartite matching’. STOC’90L. Lovasz and M. D. Plummer. `Matching Theory’. Ann Disc Math’86A. Mehta. `Online matching and ad allocation’. TCS’12.C. Swamy and D. B. Shmoys. `Approximation algorithms for 2-stage stochastic optimization problems’. SIGACT’0625

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